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Theorem pwssplit2 17119
Description: Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.b . 2  |-  B  =  ( Base `  Y
)
2 pwssplit1.c . 2  |-  C  =  ( Base `  Z
)
3 eqid 2441 . 2  |-  ( +g  `  Y )  =  ( +g  `  Y )
4 eqid 2441 . 2  |-  ( +g  `  Z )  =  ( +g  `  Z )
5 simp1 983 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Grp )
6 simp2 984 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
7 pwssplit1.y . . . 4  |-  Y  =  ( W  ^s  U )
87pwsgrp 15659 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X )  ->  Y  e.  Grp )
95, 6, 8syl2anc 656 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Y  e.  Grp )
10 simp3 985 . . . 4  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
116, 10ssexd 4436 . . 3  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
12 pwssplit1.z . . . 4  |-  Z  =  ( W  ^s  V )
1312pwsgrp 15659 . . 3  |-  ( ( W  e.  Grp  /\  V  e.  _V )  ->  Z  e.  Grp )
145, 11, 13syl2anc 656 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  Z  e.  Grp )
15 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
167, 12, 1, 2, 15pwssplit0 17117 . 2  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
17 offres 6571 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
1817adantl 463 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a  oF ( +g  `  W
) b )  |`  V )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
195adantr 462 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  W  e.  Grp )
20 simpl2 987 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  U  e.  X )
21 simprl 750 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  a  e.  B )
22 simprr 751 . . . . . 6  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  b  e.  B )
23 eqid 2441 . . . . . 6  |-  ( +g  `  W )  =  ( +g  `  W )
247, 1, 19, 20, 21, 22, 23, 3pwsplusgval 14424 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  =  ( a  oF ( +g  `  W
) b ) )
2524reseq1d 5105 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( a  oF ( +g  `  W
) b )  |`  V ) )
2615fvtresfn 5772 . . . . . 6  |-  ( a  e.  B  ->  ( F `  a )  =  ( a  |`  V ) )
2715fvtresfn 5772 . . . . . 6  |-  ( b  e.  B  ->  ( F `  b )  =  ( b  |`  V ) )
2826, 27oveqan12d 6109 . . . . 5  |-  ( ( a  e.  B  /\  b  e.  B )  ->  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) )  =  ( ( a  |`  V )  oF ( +g  `  W
) ( b  |`  V ) ) )
2928adantl 463 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
)  oF ( +g  `  W ) ( F `  b
) )  =  ( ( a  |`  V )  oF ( +g  `  W ) ( b  |`  V ) ) )
3018, 25, 293eqtr4d 2483 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( a ( +g  `  Y ) b )  |`  V )  =  ( ( F `  a
)  oF ( +g  `  W ) ( F `  b
) ) )
311, 3grpcl 15544 . . . . . 6  |-  ( ( Y  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( a ( +g  `  Y ) b )  e.  B )
32313expb 1183 . . . . 5  |-  ( ( Y  e.  Grp  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
339, 32sylan 468 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
a ( +g  `  Y
) b )  e.  B )
3415fvtresfn 5772 . . . 4  |-  ( ( a ( +g  `  Y
) b )  e.  B  ->  ( F `  ( a ( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y
) b )  |`  V ) )
3533, 34syl 16 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( a ( +g  `  Y ) b )  |`  V ) )
3611adantr 462 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  V  e.  _V )
3716ffvelrnda 5840 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  B )  ->  ( F `  a
)  e.  C )
3837adantrr 711 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  a )  e.  C )
3916ffvelrnda 5840 . . . . 5  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  b  e.  B )  ->  ( F `  b
)  e.  C )
4039adantrl 710 . . . 4  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  b )  e.  C )
4112, 2, 19, 36, 38, 40, 23, 4pwsplusgval 14424 . . 3  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  (
( F `  a
) ( +g  `  Z
) ( F `  b ) )  =  ( ( F `  a )  oF ( +g  `  W
) ( F `  b ) ) )
4230, 35, 413eqtr4d 2483 . 2  |-  ( ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  /\  ( a  e.  B  /\  b  e.  B
) )  ->  ( F `  ( a
( +g  `  Y ) b ) )  =  ( ( F `  a ) ( +g  `  Z ) ( F `
 b ) ) )
431, 2, 3, 4, 9, 14, 16, 42isghmd 15749 1  |-  ( ( W  e.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325    e. cmpt 4347    |` cres 4838   ` cfv 5415  (class class class)co 6090    oFcof 6317   Basecbs 14170   +g cplusg 14234    ^s cpws 14381   Grpcgrp 15406    GrpHom cghm 15737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-0g 14376  df-prds 14382  df-pws 14384  df-mnd 15411  df-grp 15538  df-minusg 15539  df-ghm 15738
This theorem is referenced by:  pwssplit3  17120
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